Math packages. Mathematical modeling programs

19.06.2019 Reviews

identical transformations of expressions (including simplification), analytical solution of equations and systems;

differentiation and integration, analytical and numerical;

solution of differential equations;

carrying out series of calculations with different values of initial conditions and other parameters.

At the same time, the range of tasks solved by such systems is very wide:

conducting mathematical research requiring calculations and analytical calculations;
development and analysis of algorithms;
mathematical modeling and computer experiment;
data analysis and processing;
visualization, scientific and engineering graphics;
development of graphic and calculation applications.

Principles of construction of mathematical models. The main stages of modeling.

Mathematical modeling is the creation of a mathematical description of a real object and the study of this description.

Principles of building mathematical models

Main stages of modeling

The whole modeling process can be divided into the following stages:

formulation of the modeling problem;

building a model diagram, highlighting the main parts and processes;

determining an optimization criterion or value to be calculated;

selection of the main changeable parameters;

mathematical description of the main parts and processes;

building a solution linking the variable parameters and the optimization criterion or the calculated value;

study of the solution for an extremum or calculation of the desired parameter.

Statement of the modeling problem

The problem statement is usually formulated as a verbal description. At the formulation stage, the object of modeling, the goals of building the model, and the optimization criteria should be described.

Building a model diagram, highlighting the main parts and processes

At this stage, on the basis of the problem statement, the modeling object is divided into main parts and a list of processes of interaction of these parts is determined.

This is where general-purpose packages can't help either. Specialized packages usually already contain elements of dividing the model into parts for their subject area.

A quantifiable optimization criterion or desired quantitative parameter must be formulated.

A list of all variable parameters and their characteristic quantitative expression should be formulated.

Mathematical description of the main parts and processes

The interaction of the parts of the model must be expressed by mathematical formulas. The section of mathematics that will be used for the description is chosen for reasons of convenience. Those. First of all, this section should be able to quantitatively describe this type of interaction.

The result of this stage is a system of equations or other mathematical expressions that formally describes the interaction of parts and allows a solution, i.e. dependence derivation: optimization criterion as a function of variable parameters.

In particular, it is desirable that the system of equations be closed and that a formal proof of the existence of a solution be available.

Here, only the apparatus is provided to general purpose packages. Specialized packages usually have a predefined mathematical apparatus and are based on a ready-made mathematical description of the problem.

Building a Solution Connecting Variable Parameters and an Optimization Criterion

A SOLUTION is being built, i.e. an explicit functional relationship is determined: an optimization criterion or a calculated parameter as a function of the variable parameters.

It is this stage that is the main field for applying the forces of applied packages of mathematical modeling. This is due to the fact that analytical solutions for the mathematical description of complex objects are usually impossible. And the construction of the solution is reduced to the construction of a "numerical solver", which, according to the given values of the variable parameters, can calculate the value of the optimization criterion.

In rare cases of the existence of an analytical solution of the model, the role of applied packages of mathematical modeling is reduced to the definition of a solution function.

There are special subsystems of applied packages of mathematical modeling - systems of analytical (symbolic) calculations - these subsystems can be used to maximize the analyticity of the solution, i.e. replacement of numerical methods with the search for a functional expression of solutions. Analytical solutions are almost always “better” than numerical ones, because they allow expressing the desired patterns in terms of known functions, which greatly speeds up calculations and increases the accuracy of calculations.

Investigation of a solution to an extremum

The complexity of investigating the solution to the extremum is most often associated with the significant time spent on calculating the optimization criterion for the given values of the variable parameters and/or the large number of acceptable combinations of the variable parameters, which leads to a huge number of calculations and, again, a significant amount of time.

This stage is another field for applying forces to packages. Methods for studying functions for extrema are well developed in mathematics and can be formally applied to any given function.

Parametric Surface Creator

Surfer

package Simulink

gnuplot ImageMagick

Parametric Surface Creator

The program is designed for visual representation of geometric objects described by parametrically defined surfaces, such as sphere, torus, Möbius strip and others. To describe objects, a Pascal-like language is used with support for all standard mathematical functions of the Pascal language and several additional ones. The resulting object is displayed in vector form using the original vector rasterization algorithm, which allows you to get a smooth and natural image even at low monitor resolutions and does not require any hardware support. It is possible to export the image to a BMP file.

Surfer- a program for creating three-dimensional surfaces. Commercial simulation programs for tasks with a predominance of "logical aspects": AutoMod, Process Model, SIMFACTORY, etc.

package Simulink, focused specifically on the tasks of simulation modeling.

gnuplot 1 is a popular program for creating two- and three-dimensional graphs. gnuplot has its own command system, can work interactively (in command line mode) and execute scripts read from files. Used by gnuplot as an image output system in various math packages: GNU Octave, Maxima, and many others. ImageMagick is a cross-platform software package for batch processing of graphic files. Supports a huge number of graphic formats. Can be used with Perl, C, C++, Python, Ruby, PHP, Pascal, Java languages, in shell scripts or on its own.

Using Components

Mathcad program documents have the ability to insert modules (component

) other applications to expand the possibilities of visualization, data analysis, performance of specific calculations.

The Axum Graph component is designed for advanced data visualization. To work with tabular data - Microsoft Excel.

Data Acquisition Components, ODBC Input allow you to use external databases.

There are also free modules (add-in) for integrating Mathcad with Excel programs, AutoCAD.

The Axum S-PLUS Script component is intended for statistical analysis.

A significant expansion of the package's capabilities is achieved by integrating with the super-powerful MATLAB application.

Complete set

Versions of Mathcad may differ in package contents and user license. Versions were delivered at different times Mathcad Professional, Mathcad Premium, Mathcad Enterprise Edition(differ in configuration). For academic users, the version is intended Mathcad Academic Professor(has full functionality, but differs in a user license and has a several times lower cost).

For some time, simplified and noticeably “cut down” student versions of the program were also produced.

However, while the mathematical capabilities of MathCad in the field of computer algebra are much inferior to Maple, Mathematica, MatLab, and even the little Derive. However, many books and training courses have been published under the MathCad program, including in Russia. Today, this system has become literally the international standard for technical computing, and even many schoolchildren master and use MathCad. For a small amount of calculations, MathCad is ideal - here everything can be done very quickly and efficiently, and then format the work in the usual way (MathCad provides ample opportunities for formatting the results, up to publication on the Internet). The package has convenient data import/export capabilities. For example, you can work with Microsoft Excel spreadsheets right inside a MathCad document.

In general, MathCad is a very simple and convenient program that can be recommended to a wide range of users, including those who are not very knowledgeable in mathematics, and especially to those who are just learning its basics.

As cheaper, simple, but ideologically close alternatives to the MathCad program, one can note such packages as the already mentioned YaCaS, the commercial MuPAD system ( http://www.mupad.de/) and the free KmPlot program

Mupad math package

As for the MuPAD program (Figure 2.6), it is a modern integrated system of mathematical calculations, with which you can perform numerical and symbolic transformations, as well as draw two-dimensional and three-dimensional graphs of geometric objects. However, in terms of its capabilities, MuPAD is significantly inferior to its venerable competitors and is rather an entry-level system designed for training.

MuPAD Pro 3 is a relatively new computer algebra system with an extensive set of tools, including mathematical algorithms for symbolic and numerical calculations, and tools for visualization, animation and interactive manipulation of 2D and 3D graphs and other mathematical objects.

Key features of Matlab

Platform-independent high-level programming language focused on matrix calculations and algorithm development

Interactive environment for code development, file and data management

· Linear algebra functions, statistics, Fourier analysis, solution of differential equations, etc.

· Rich visualization tools, 2-D and 3-D graphics.

Built-in user interface development tools for creating complete MATLAB applications

C/C++ integration tools, code inheritance, ActiveX technologies

The basic set of MatLab includes arithmetic, algebraic, trigonometric and some special functions, fast direct and inverse Fourier transform and digital filtering functions, vector and matrix functions. MatLab "can" perform operations with polynomials and complex numbers, build graphs in Cartesian and polar coordinate systems, form images of three-dimensional surfaces. MatLab has tools for calculating and designing analog and digital filters, building their frequency, impulse and transient characteristics and the same characteristics for linear electrical circuits, tools for spectral analysis and synthesis.

The C Math library (MatLab compiler) is an object library and contains over 300 data processing procedures in the C language. Inside the package, you can use both the procedures of the MatLab itself and the standard procedures of the C language, which makes this tool a powerful help when developing applications (using the C Math compiler , you can embed any MatLab procedures in ready-made applications).

The C Math library allows you to use the following categories of functions:

operations with matrices;

comparison of matrices;

solving linear equations;

decomposition of operators and search for eigenvalues;

finding the inverse matrix;

search for a determinant;

calculation of the matrix exponential;

elementary mathematics;

functions beta, gamma, erf and elliptic functions;

basics of statistics and data analysis;

search for the roots of polynomials;

filtering, convolution;

Fast Fourier Transform (FFT);

· interpolation;

Operations with strings

· file I/O operations, etc.

At the same time, all MatLab libraries are characterized by high speed of numerical calculations. However, matrices are widely used not only in such mathematical calculations as solving problems of linear algebra and mathematical modeling, calculation of static and dynamic systems and objects. They are the basis for the automatic compilation and solution of the equations of state of dynamic objects and systems. It is the universality of the matrix calculus apparatus that significantly increases the interest in the MatLab system, which incorporates the best achievements in the field of fast solution of matrix problems. Therefore, MatLab has long gone beyond the specialized matrix system, turning into one of the most powerful universal integrated systems of computer mathematics.

Maple math package.

maple( http://www.maplesoft.com/)

Processor Pentium III 650 MHz;

400 MB disk space;

Operating systems: Windows NT 4 (SP5)/98/ME/2000/2003 Server/XP Pro/XP Home.

The Maple program (latest version 10.02) is a kind of patriarch in the family of symbolic mathematics systems and is still one of the leaders among universal symbolic computing systems. (Figure 2.15,2.16) It provides the user with a convenient intellectual environment for mathematical research at any level and is especially popular in the scientific community.

Note that the symbolic analyzer of the Maple program is the most powerful part of this software, so it was borrowed and included in a number of other CAE packages, such as MathCad and MatLab, as well as in Scientific WorkPlace and Math Office for Word packages for preparing scientific publications. . The Maple package is a joint development of the University of Waterloo (Ontario, Canada) and the Higher Technical School (ETHZ, Zurich, Switzerland).

For its sale, a special company was created - Waterloo Maple, Inc., which, unfortunately, became more famous for the mathematical elaboration of its project than for the level of its commercial implementation. As a result, the Maple system was previously available mainly to a narrow circle of professionals. Now this company works together with the more successful in commerce and in the development of the user interface of mathematical systems, MathSoft, Inc. - the creator of very popular and mass systems for numerical calculations MathCad, which have become the international standard for technical calculations.

Maple provides a convenient environment for computer experiments, during which various approaches to the problem are tried, particular solutions are analyzed, and, if necessary, programming fragments that require special speed are selected.

The package allows you to create integrated environments with the participation of other systems and high-level universal programming languages. When the calculations are made and it is required to formalize the results, then you can use the tools of this package to visualize the data and prepare illustrations for publication. To complete the work, it remains to prepare printed material (report, article, book) directly in the Maple environment, and then you can proceed to the next study. The work is interactive - the user enters commands and immediately sees the result of their execution on the screen. At the same time, the Maple package is not at all like a traditional programming environment, where a strict formalization of all variables and actions with them is required. Here, the choice of appropriate types of variables is automatically ensured and the correctness of the operations is checked, so that in the general case there is no need for a description of variables and strict formalization of the notation.

The Maple package consists of a core (procedures written in C and well optimized), a library written in the Maple language, and a rich front-end. The kernel performs most of the basic operations, and the library contains many commands - procedures that are executed in interpretation mode.

The Maple interface is based on the concept of a worksheet, or document containing I/O lines and text, as well as graphics (Figure 2.17).

The package is processed in the interpreter mode. In the input line, the user specifies a command, presses the Enter key and receives the result - an output line (or lines) or a message about an erroneously entered command. An invitation to enter a new command is immediately issued, etc.

Calculations in Maple

The Maple system can be used at the most elementary level of its capabilities - as a very powerful calculator for calculating given formulas, but its main advantage is the ability to perform arithmetic operations in symbolic form, that is, the way a person does it. When working with fractions and roots, the program does not convert them to decimal form during calculations, but makes the necessary reductions and conversions to a column, which allows you to avoid rounding errors.

To work with decimal equivalents, the Maple system has a special command that approximates the value of an expression in floating point format. The Maple system calculates finite and infinite sums and products, performs computational operations with complex numbers, easily converts a complex number to a number in polar coordinates, calculates the numerical values of elementary functions, and also knows many special functions and mathematical constants (such as "e " and "pi"). Maple supports hundreds of special functions and numbers found in many areas of mathematics, science and technology.

Programming in Maple.

The Maple system uses the 4th generation procedural language (4GL). This language is specifically designed for the rapid development of mathematical routines and custom applications. The syntax of this language is similar to the syntax of the high-level universal languages: C, Fortran, Basic and Pascal.

Maple can generate code that is compatible with programming languages such as Fortran or C, and with the LaTeX typing language, which is very popular in the scientific world and is used for publishing. One of the advantages of this property is the ability to provide access to specialized numerical programs that maximize the speed of solving complex problems. For example, using the Maple system, you can develop a certain mathematical model, and then use it to generate C code corresponding to this model. The 4GL language, specially optimized for developing mathematical applications, allows you to shorten the development process, and Maplets elements or Maple documents with integrated graphical components help you customize the user interface.

At the same time, in the Maple environment, you can also prepare documentation for the application, since the package tools allow you to create professional-looking technical documents containing text, interactive mathematical calculations, graphics, drawings, and even sound. You can also create interactive documents and presentations by adding buttons, sliders, and other components, and finally publish documents on the Web and deploy interactive computing on the Web using the MapleNet server.

Mathematica package.

Mathematica ( http://www.wolfram.com/)

Minimum system requirements:

Pentium II processor or higher;

400-550 MB disk space;

operating systems: Windows 98/Me/NT 4.0/2000/2003 Server/2003x64/XP/XP x64.

Wolfram Research, Inc., which developed the Mathematica computer mathematics system (Figure 2.27,2.28), is rightfully considered the oldest and most solid player in this area. The Mathematica package (current version 5.2) is widely used in calculations in modern scientific research and has become widely known in the scientific and educational environment. You can even say that Mathematica has a significant functional redundancy (there, in particular, there is even an opportunity for sound synthesis).

Mathematica combines a numerical and symbolic computing core, a graphics system, a programming language, a documentation system, and the ability to interact with other applications into a single whole. For the entire Mathematica environment, there is no single competitor. Broadly speaking, competitors fall into the following groups: numerical packages, computer algebra systems, typing and documentation applications, graphics and statistical systems, traditional programming languages (interface development tools), and spreadsheets. Since Mathematica was first introduced, other math packages have greatly expanded their range of capabilities, originally designed to solve problems that fall into just one or two of the above categories.
However, it is unlikely that this powerful mathematical system, which claims to be world leader, is needed by a secretary or even the director of a small commercial company, not to mention ordinary users. But, undoubtedly, any serious scientific laboratory or university department should have such a program if they are seriously interested in automating the performance of mathematical calculations of any degree of complexity. Despite their focus on serious mathematical calculations, Mathematica class systems are easy to learn and can be used by a fairly wide category of users - university students and teachers, engineers, graduate students, scientists, and even students of mathematical classes in general education and special schools. All of them will find numerous useful applications in such a system.

At the same time, the broadest functions of the program do not overload its interface and do not slow down calculations. Mathematica consistently demonstrates the high speed of symbolic conversions and numerical calculations. Of all the systems under consideration, Mathematica is the most complete and versatile, but each program has its own advantages and disadvantages. And most importantly, they have their adherents, whom it is useless to convince of the superiority of another system. But those who work seriously with computer mathematics systems should use several programs, because only this guarantees a high level of reliability of complex calculations.

Note that in the development of various versions of the Mathematica system, along with the parent company Wolfram Research, Inc., other companies and hundreds of highly qualified specialists, including mathematicians and programmers, took part. Among them are representatives of the Russian mathematical school, which is respected and in demand abroad. The Mathematica system is one of the largest software systems and implements the most efficient calculation algorithms. Among them, for example, is the mechanism of contexts, which excludes the appearance of side effects in programs.

Mathematica is now regarded as the world's leading symbolic mathematics computer system for the PC, providing not only the ability to perform complex numerical calculations with the output of their results in the most sophisticated graphical form, but also the performance of particularly time-consuming analytical transformations and calculations.

Mathematica has several main features and is designed to solve a wide range of problems. Here are some classes of problems solved with Mathematica:

1. Working with symbolic complex calculations using hundreds of thousands or millions of members.
Loading, analysis and visualization of data.

2. Solution of ordinary and differential equations, as well as problems of numerical or symbolic minimization.

3. Numerical modeling and simulation, building control systems, ranging from the simplest to collisions of galaxies, financial losses, complex biological systems, chemical reactions, studying the impact on the environment and magnetic fields in particle accelerators.

4. Easy and fast application development (RAD) for tech companies and financial institutions.

5. Create professional, interactive, technical reports and documents for distribution electronically or on paper.

6. Detailed technical documentation, eg for US patents.

7. Conducting special presentations and seminars.

8. Illustrate math or science concepts for students from college to graduate school.

Versions of the system under Windows have a modern user interface and allow you to prepare documents in the form of Notebooks (notebooks). They combine source data, descriptions of algorithms for solving problems, programs and solution results in a wide variety of forms (mathematical formulas, numbers, vectors, matrices, tables and graphs).

Mathematica was conceived as a system that automates the work of scientists and analytical mathematicians as much as possible, so it deserves to be studied even as a typical representative of elite and highly intelligent software products of the highest degree of complexity. However, it is of much greater interest as a powerful and flexible mathematical toolkit that can provide invaluable assistance to most scientists, university and university professors, students, engineers, and even schoolchildren.

From the very beginning, much attention was paid to graphics, including dynamic ones, and even multimedia capabilities - dynamic animation playback and sound synthesis. The set of graphics functions and options that change their action is very wide. Graphics has always been a strong point of the various versions of Mathematica and has given them the lead among computer mathematics systems.

As a result, Mathematica quickly took a leading position in the market for symbolic mathematical systems. Particularly attractive are the extensive graphical capabilities of the system and the implementation of the Notebook-type interface. At the same time, the system provided a dynamic connection between the cells of documents in the style of spreadsheets, even when solving symbolic tasks, which fundamentally and favorably distinguished it from other similar systems.

By the way, the central place in systems of the Mathematica class is occupied by a machine-independent core of mathematical operations, which allows you to transfer the system to various computer platforms. To transfer the system to another computer platform, the Front End software interface processor is used. It is he who determines what kind of user interface the system has, that is, the interface processors of Mathematica systems for other platforms may have their own nuances. The kernel is made compact enough to be able to call any function from it very quickly. To expand the set of functions, a library (Library) and a set of extension packages (Add-on Packages) are used. Extension packages are prepared in Mathematica's own system programming language and are the main means for developing the system's capabilities and adapting them to solving specific classes of user problems. In addition, the systems have a built-in electronic help system - Help, which contains electronic books with real examples.

Thus, Mathematica is, on the one hand, a typical programming system based on one of the most powerful high-level problem-oriented functional programming languages, designed to solve various problems (including mathematical ones), and on the other hand, an interactive system for solving most mathematical problems. tasks interactively without traditional programming. Thus, Mathematica as a programming system has all the possibilities for developing and creating almost any control structures, organizing I / O, working with system functions and servicing any peripheral devices, and with the help of extension packages (Add-ons), it becomes possible to adapt to the needs of any user (although an ordinary user may not need these programming tools - he will completely manage with the built-in mathematical functions of the system, which amaze even experienced mathematicians with their abundance and variety).

The disadvantages of the Mathematica system include perhaps a very unusual programming language, which, however, is facilitated by a detailed help system.

FlatGraph is a program for constructing graphs of functions (normal and parametric) with advanced features (Figure 2.33). Differentiation of any order (with simplification). Construction of tangents to the graph. The program is designed for both inexperienced and professional users, because it combines an intuitive interface with professional features.

FlatGraph allows you to:

Enter one or more functional expressions of any complexity for display and (or) their differentiation;

Perform symbolic differentiation for the specified order of the derivative, as well as perform simplification of the resulting derivative;

Explore the "live" change of various function parameters with the simultaneous display of new graphs, which allows you to determine the effect of function parameters on their appearance;

Use automatic or manual scaling of function graphs for linear scales;

Set and display graphically parametric functions, displaying, for example, ellipsoids, cardioid, Bernoulli's lemniscates and other similar graphs (where the abscissa and ordinate depend on one parameter "t");

Solve equations, systems of equations and inequalities graphically;

Get and display the tangent to the graph of the function at the point x0 (set by the user).

FlatGraph has a simple and intuitive interface, provided with detailed documentation on how to use it and examples of how it works.

Math packages. Modeling. List the features and main tasks solved by packages.

Math packages are an integral part of the world of CAE-systems. (Computer Aided Engineering) Nowadays, math packages apply the principle of model construction, and not the traditional "art of programming". That is, the user sets a task, and the system finds methods and algorithms for solving it itself. Modern mathematical packages can be used both as a regular calculator and as a means to simplify expressions when solving any problems, as well as a graphics or even sound generator! At present, almost all modern mathematical programs have built-in functions for symbolic calculations. However, Maple, MathCad, Mathematica and MatLab are considered the most famous and adapted for mathematical symbolic calculations. Mathematical modeling - creation of a mathematical description of a real object and the study of this description.

Initially, any calculations on the models were made manually. As computing devices evolved, these devices were used to speed up calculations.

The computer allows using it as a means of automating scientific work and various specialized programs are used to solve complex calculation problems.

At the same time, in scientific work there is a wide range of simple mathematical problems, for which universal professional tools can be used.

Such simple tasks include, for example, the following:

preparation of scientific and technical documents containing text and formulas written in a form familiar to specialists;

calculation of the results of mathematical operations involving numerical constants, variables and dimensional physical quantities;

operations with vectors and matrices;

solution of equations and systems of equations (inequalities);

statistical calculations and data analysis;

construction of two-dimensional and three-dimensional graphs;

identical transformations of expressions (including simplification), analytical

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Introduction

Today, computers take on a huge share of the computational and analytical workload of a modern mathematician. Therefore, today's researchers face and, most importantly, completely different tasks than half a century ago seem to be solvable.

Thanks to the enormous power of computers, it becomes possible to model and study complex and dynamic systems that arise in the study of space, the search for new energy sources, the creation of new technical inventions and many other problems affecting the sphere of scientific and technological progress. The solution of any problem of this kind can be reduced to the following set of actions:

· mathematical modeling of the system;

construction of a computational algorithm;

carrying out calculations;

collection and analysis of the obtained results.

Leading mathematical packages now, with minimal familiarity, easily carry out very complex analytical transformations of mathematical expressions, take derivatives, integrals, calculate limits, expand and simplify expressions, draw graphs. And now you do not need to study programming languages for a long time to master the mathematical capabilities of a computer. Now almost everything necessary for an engineer, economist, sociologist, statistician is implemented in mathematical packages. Such world-famous packages as Mathematica, Mathcad, MatLAB, Maple, have become not only a convenient computing, but also an amazingly fruitful, flexible educational environment. In my opinion, together with the Internet, these packages can unite the efforts of many, many people, providing powerful educational initiatives. Indeed, in computer textbooks and lectures, not ordinary, but directly executable formulas are now inserted into the text, with the help of which the essence of phenomena is clearly demonstrated. They can be modified for their own tasks, supplemented and expanded, resulting in not only numbers, but also new analytical expressions, graphs, tables.

The use of computer mathematical packages allows:

expand the range of real applications;

· for visual analysis, build graphs of complex functions and surfaces, with the help of which, for example, ODE solutions are estimated, which greatly facilitates their analysis;

· to combine professional orientation, scientific character, consistency, visualization, interactivity, interdisciplinary connections in solving ODE;

Instantly exchange information with a person with whom physical contact is impossible or difficult to implement;

Consider more tasks by reducing the number of routine transformations;

Investigate more complex models, since cumbersome calculations can be carried out using appropriate computer systems;

Pay more attention to the qualitative aspects of your task.

The purpose of this work is the use of information technology for mathematical calculations on the example of the Maple package.

1. Study the literature on this topic.

2. Conduct a comparative analysis of modern mathematical packages: Mathematica, Maple V, MatLAB, Derive, Mathcad.

3. Apply the Maple package in mathematics lessons.

4. Make a conclusion on the work done.

1. Modern mathematical packages in education

1.1 conceptand usemath packagesin education

Methods and forms of application of computer technologies in the educational process is an actual methodological and organizational task of every teacher, every administrator of a school, university.

When organizing computer support for education, two directions can be distinguished:

development of computer programs for educational purposes, programs specially designed for the study of a particular discipline;

use of software developed for professional activities in the relevant field of knowledge; for most natural science disciplines, these are professional mathematical packages.

Mathematical packages here are called systems, environments, languages such as Mathematica, Maple V, MatLAB, Derive, Mathcad, as well as a family of statistical data analysis systems - such as SPSS, Statistica, Statgraphics, Stadia, etc. Modern mathematical packages are programs (software packages ), which have the means to perform various numerical and analytical (symbolic) mathematical calculations, from simple arithmetic calculations to solving partial differential equations, solving optimization problems, testing statistical hypotheses, tools for constructing mathematical models and other tools necessary for performing various technical calculations. All of them have advanced scientific graphics, a convenient help system, as well as reporting tools. The name "professional" or "universal" is used as an alternative to the name "training package".

For many years, mathematics teachers were quite clearly divided into supporters of the use of computer programs for educational purposes ("educational packages", training programs) and those who preferred to use universal packages.

There are several key points that determined the fundamental change in the attitude of teachers and students to the use of universal mathematical packages.

The computer has become an element of "home appliances". The modern concept of quality education includes fluency in computer technologies as a necessary element and, as a result, the computer is perceived as an object, if not the first, then the second necessity. Most parents cannot imagine raising their own schoolchildren without a computer. An increasing number of students have computers at home and more and more often it is students who initiate the use of computer technology in the educational process. They are not driven by a "gaming" interest, as we said and saw before, but by the desire to "make life easier", the desire to acquire professional skills useful for a future career, and the willingness to learn how to work on a computer not only in special classes in computer science. We can safely say that the "home computer" is the most powerful factor that has changed the attitude of teachers to the use of a computer in their professional activities. Their position is changing under the influence of public opinion, under the influence of the position of students, and also because many teachers also have computers at home. This explains the interest in universal packages - learning to work with ready-made software is much easier than writing programs yourself.

In the modern world, standards have been formed and consolidated in the organization of the interface of computer programs. One of the problems that arise when using universal packages is the cost of study time for studying the rules of working with the program (for studying the interface). However, since the developers of scientific software and the developers of "mass consumption" packages adhere to the same standards. Thanks to this, the time to learn the interface of a particular scientific package is reduced by using the skills of working with office programs.

The struggle for the consumer, the desire to expand the circle of users, has led to the fact that while maintaining individual features, the packages converge, become so similar that the skills of working with one of them allow you to quickly get used to working in any other. Developers of mathematical packages very quickly equip their programs with all technological innovations, quickly release versions for new platforms and operating systems, improve command languages, incorporating the latest achievements in algorithmic languages, etc. The intellectual capabilities of packages are developing: new libraries, modules are added, the range of tasks available for research is expanding in accordance with fashion, with the advent of new applications, new research methods, etc.

The Internet is a new reality in the life of a modern student and specialist. Thanks to global computer networks, the user of any common software product gets the opportunity to join the global community of consumers of the same product. He will find information about new products on the net, the latest versions of the program, messages about detected errors, get expert advice, talk about his findings and get acquainted with the tricks of others, learn about literature, about the range of problems to be solved, often just find a solution to a similar problem, etc. P.

A separate place is occupied by statistical packages. Today, mathematical statistics is by far the most demanded mathematical course. The methods of data analysis studied here are widely used in practice. Therefore, mastering the methods of working in the environment of a universal statistical package is an element of high-quality professional education that is in demand on the labor market.

Mathematical packages - a tool for educational activities. A university student works, his work is study. The more perfect the tools that the student uses, the higher the results he achieves. The use of mathematical packages simplifies the preparation of reports on laboratory work, helps to overcome technical mathematical difficulties in solving engineering problems, expands the range of problems available for solving, and helps to present the results of calculations in a visual graphical form. If already in the junior years, when studying mathematics, physics, biology, a student masters the techniques of working with a fairly powerful professional package, then he is much better prepared for solving mathematical problems in various applications. He will not be afraid of cumbersome calculations, will be ready to solve complex problems, compensating for the lack of his own knowledge using the intellectual capabilities of the package, has the skills to present research results in a visual graphical form, and is able to draw up research results in the form of accurate meaningful reports.

Availability of universal mathematical packages and their on the professional software market. An essential circumstance that until recently prevented the widespread use of professional packages within the walls of universities is the high cost of professional scientific mathematical software. Recently, however, many firms developing and distributing programs for science present for free use (including via global networks) previous versions of their programs, widely use the system of discounts for educational institutions, and distribute demo or short-lived versions for free. Publicly available, freely distributed versions of the packages contain the main computational and graphical tools and, therefore, are quite suitable for use in the educational process (modernization of mathematical packages is carried out mainly in the direction of expanding the range of tasks available for professional research by adding more and more subtle computational methods, expanding the capabilities of command languages and adapting to the latest advances in information technology). On the other hand, the use of high-quality software contributes to the intensification of research activities, allows students to be more widely involved in scientific work, which, as is known, improves the chances of scientific groups in the distribution of grants, and, therefore, allows them to later find funds to purchase more modern licensed software. .

Availability of documentation and reference literature on mathematical packages. If relatively recently there was practically no literature on packages in Russian, now new versions, new packages and various user manuals for them appear almost simultaneously. It is difficult to find a package that would not come out in Russian for two or three books.

It should be noted that developers willingly provide authors with proprietary documentation and the latest versions of packages for work. In addition, almost all developers maintain servers that host descriptions of the latest innovations, information about detected errors, extended guides for working with the package, descriptions of examples of solving typical problems, and, almost always, information about users in the academic environment with addresses, descriptions of experience and examples of use in education. It can be stated that today the reference literature on mathematical packages is publicly available - any user who wants to get acquainted with this or that package and learn how to work with it has the opportunity to get help that meets his personal needs and qualifications.

1.2 Comparative analysis of Au math packagestoCad, MatLab, Maple, Mathematica

The analysis consists of a table that lists the functionality of the programs. It is divided into functional sections of mathematical, graphical, functional capabilities and in the programming environment, a section on data import / export, the possibility of using it in various operating systems, a comparison of speed and information in general. To simplify the analysis of all data, we used a simple scoring system.

A score of 1 was given to those programs that have automatic functions, a score of 0.9 is given to those applications that need to be installed separately. Programs in which automatic functions are not available receive a score of 0 points. The sum in each column is the total score.

As a result, all scores were evaluated as follows:

Math functions 38%;

Graphic functions 10%;

Programming software 9%;

Data import/export 5%;

Operating systems 2%;

Speed comparison 36%.

Common symbols used in various schemes

The function is built into the program

m - The function is supported by an additional module, which can be downloaded for free.

$ - The function is supported by an additional module, which can be downloaded for a fee.

The features listed are all based on commercial products (except Scilab) that have warranties and support. Of course, there are a huge number of free software applications, modules available, but no guarantee of service or support. This is a very important item for several types of activities (ie bank use).

Comparison of mathematical functionality

In fact, there are many different mathematical and statistical programs on the market that cover a huge number of functions.

The following table should give an overview of the functionality for analyzing data in numerical ways and should indicate which functions are supported by which programs, whether these functions are already implemented in the main program or if you need an additional module.

Algebra and especially linear algebra offer basic functionality for any kind of oriented matrix operation. That is, the types of optimization widely used in the financial sector are also very useful in comparing speed.

The following speed comparison was made on a Pentium-III with a 550 MHz processor and 384 MB RAM running under Windows XP. Since one would expect that modern computers could solve these problems within a short time, the maximum duration for each function was limited to 10 minutes.

The Speed Comparison tests 18 functions that are very commonly used in math models. It is necessary to interpret the timing results in content with whole models as then small differences in the timings of single functions might results in timing differences of minutes to several hours. However, it is not possible to use full models for these evaluation tests as a job for making the model work in every math package, and also the duration would be very high.

Functions (Version)

Reading data from an ASCII data file
Reading data from the database via the ODBC interface
Extracting a descriptive statistic
Loop test 5000 x 5000
3800x3800 random matrix^1000
Sorting 3,000,000 random values
FFT over 1048576 (= 2^20) random values
Triple Integration
Determinant 1000x1000 random matrix
Invert 1000x1000 random matrix
Eigenvalues 600x600 random matrix
Cholesky decomposition 1000x1000 random matrix
1000x1000 crossproduct matrix
Computing 1000000 Fibonacci Numbers
Main component factorization by 500x500 matrix
Gamma function on 1500x1500 random matrix
Gaussian error function on 1500x1500 random matrix
Linear regression over 1000x1000 random matrix
Full work

* - The maximum duration of 10 minutes has been exceeded.

The total work was calculated as follows:

The best performance result of the function is estimated as 100%; for calculating the results for each function I'll take the best performance and divide it by the tested program's timing (the formula will look like MIN(A1;A2;...)/A2) and this is displayed as a percentage. To make the final "Full Job" I will calculate the sum of the percentages and divide by the number of programs, which is again displayed as a percentage.

Features that are not supported by the program will not be evaluated.

General information about the product.

Some amount of information like pricing, support, newsgroups, books, etc. are essential to users of mathematical or statistical software. Due to the fact that this type of information cannot be characterized objectively, one can only mention them without judgment for the final summary of the test report.

Functions (Version)

Operation / Programming processing
User interface

Programming language (similar)			(Basic, Fortran)
Online help / Electron. management
Add. books
FAQ Lists
Teleconferences / mailing lists
Program archives by software maker
Program archives by external institutions

The information in this table is rated on a scale of 1 to 6 (1 being best, 6 being worst) and represents my own subjective opinion. A score of 6 usually means that something is not supported, which means that this feature is really badly supported. A score of 1 is given to the feature that is best supported.

Miscellaneous information: The summary should establish the comparison results of speed, functionality of the software environment, data import/export services and suitability for various platforms with respect to the results of comparison of mathematical and graphical functionality. The ratio between these four tests is 38:10:9:5:2:36.

Functions (Version)

Comparison of math functionality (38%)
Comparison of graphical functionality (10%)
Software environment functionality (9%)
Data circulating (with 5%)
Available Platforms (2%)
Speed Comparison (36%)
Full result

Summary: The overall results of some tested programs are not the best due to a certain overhead of this test report.

2. Development of programming skills among schoolchildren in the environmentmaple

2.1 The concept of programmatic development of a library of procedures in the environmentmaple

The Maple package consists of a fast kernel written in C containing basic mathematical functions and commands, as well as a large number of libraries that extend its capabilities in various areas of mathematics. The libraries are compiled from subroutines written in Maple's own language, specifically designed for creating symbolic computation programs. The most interesting features of the Maple system are the editing and modification of these subroutines, as well as the addition of subroutines designed to solve specific problems to libraries. They have already appeared in large numbers, and the best of them are included in the Share-library of users distributed with the Maple package.

The program has already turned into a powerful computing system that allows you to perform complex algebraic transformations, including over the field of complex numbers, calculate finite and infinite sums, products, limits and integrals, find the roots of polynomials, solve analytically and numerically algebraic (including transcendental) systems equations and inequalities, as well as systems of ordinary differential equations and partial differential equations. Maple includes specialized subroutine packages for solving problems of analytical geometry, linear and tensor algebra, number theory, combinatorics, probability theory and mathematical statistics, group theory, numerical approximation and linear optimization (simplex method), financial mathematics, integral transformations, etc. P.

Creating a new library is as follows.

First of all, you need to determine the name of your library, for example mylib, and create a directory (folder) for it on disk with the given name. Procedures in Maple are associated with tables. Therefore, first you need to set a dummy table for future procedures:

> mylib:=tab1e():

mylib:=table()

Now we need to enter our library procedures. They are specified with a double name - first the name of the library, and then the name of the procedure in square brackets. For example, let's define three simple procedures named fl, f2 and f3:

> mylib:=proc(x: Anything) sin(x)+cos(x) end:

> mylib:=proc(x:anything) sin(x)^2+cos(x)^2 end:

> mylib:=proc(x::anything) if x=0 then 1 else sin(x)/x fi end:

You can plot the introduced procedure-functions. They are represented in the with function to verify that the mylib library actually contains the procedures that have just been introduced into it. Their list should appear when calling with (mylib):

>with(mylib);

Now you need to write this library under your name to disk using the save command:

> save(mylib,`c:/mylib.m);

Pay special attention to the correct specification of the full file name. The \ character, commonly used to indicate a path, is used as a line continuation character in Maple language strings. Therefore, either the double \\ sign or the / sign must be used. In this example, the file is written to the root of drive C. It is better to place the library file in another folder (for example, in a library that already exists in the system), the full path to it is indicated.

After all this, you need to make sure that the library file is written. After that, you can immediately and count it. To do this, you must first eliminate the previously introduced procedure definitions with the restart command:

With the with command, you can verify that these definitions are no longer there:

> with(mylib):

Error, (in pacman:-pexports) mylib is not a package

After that, with the read command, you need to load the library file:

> read("c:/mylib.m");

The file name must be specified according to the rules specified for the save command. If everything is done punctually, then the with command should show the presence of a list of procedures fl, f2 and f3 in your library:

> with(mylib):

And finally, you can try again the work of the procedures, which are now introduced from the loaded library:

sin(x) + cos(x) > simplify(f2(y));

The method of creating your own library described above will suit most users. However, there is a more complex and more "advanced" way to add your own library to the existing one. To implement this, Maple has the following operations for writing to the library of procedures si, s2, ... and reading them from files filel, file2, ...:

savelib(s1. s2, .... sn, filename)

readlib(f. file1. file2. ...)

You can use the special makehelp statement to provide a standard help description for new procedures:

makehelp(n.f.b).

where n is the name of the topic, f is the name of the text file containing the help text (the file is prepared as a Maple document), and b is the name of the library. The libname system variable holds the name of the directory of library files. To register the created certificate, you need to execute a command of the form:

libname:-libname. "/mylib":

See the help system for details on how to use these operators. mathematical programming calculation maple

You should be very careful when creating your own library procedures. Using them renders your Maple programs incompatible with the standard version of Maple. If you use one or two procedures, it's easier to put them in the documents that really need them. Otherwise, you will be forced to attach a library of procedures to each of your programs. It often turns out to be larger in size than the file of the document itself. It is not always practical to attach a small document file to a large library, most of whose procedures are most likely simply not needed for this document. It is especially risky to change the Maple standard library.

However, whether to go for it or not is up to each user. Of course, if you create a serious library of your procedures, then it must be written down and carefully stored. Maple comes with many libraries of useful routines compiled by users from all over the world, so you can add your own creations to it.

2.2 Programmatic development of a library of procedures in the environmentmaple- as a factor in the development of programming skills

From the experience of some schools, it became known that in recent years there has been a constant reduction in teaching hours in the subjects of the physical and mathematical cycle with a simultaneous expansion of the list of issues studied. In this regard, there was a need for an additional and effective study of such basic subjects as mathematics, physics and computer science, as well as other disciplines of the natural science cycle. The idea of integrating these disciplines is undoubtedly very productive, since, on the one hand, it provides a basis for studying these subjects, and on the other hand, it allows developing an information and mathematical culture in the learning process and instilling applied research skills. At the same time, information technology can provide the necessary tools for this integration. In particular, the Maple computer mathematics system is considered as one of such tools.

In practice, one of the schools implemented the program "Integration of physical and mathematical education based on information technology and the package of symbolic mathematics Maple".

The program involved 10-11 classes of information technology and physical and mathematical profiles. The study of the capabilities of the Maple symbolic mathematics package and its subsequent application was of an applied nature: students of the physics and mathematics class expanded and deepened their knowledge of mathematics, got the opportunity to visualize various mathematical situations, and the classes of the information technology profile received useful professional skills as programmers and computer operators . During the implementation of the concept of specialized education at the senior level, it was especially relevant to introduce into the process of teaching computer science and information technology such systems and programs that enable students to reveal their mental and creative abilities, gain basic professional skills and determine the course of their future career. Also, students needed to instill the skills and abilities of computer modeling, which was one of the priority areas in applied sciences.

The experience of using computer mathematics both in universities and at school indicates that of the well-known mathematical packages, Maple is optimal for educational purposes. A number of features of Maple put it in a leading position for the implementation of educational programs: a relatively low cost of the package, a simple and understandable interface, a programming language closest to the language of mathematical logic, and unsurpassed graphical capabilities. All these features make it possible to present the mathematical model of the object or phenomenon under study in a visual interactive graphical form, thereby significantly improving the quality of projects in physical and mathematical disciplines. It is important to note that the results obtained, including animation models of objects and processes, are easily exported to Web pages and text documents.

The introduction of Maple into the education system is carried out in the form of conducting an elective course "Studying the package of symbolic mathematics Maple" (grade 11), the main task of which is to create the necessary conditions for the implementation of the experiment program. The main goal of the experimental work on the introduction of Maple into the learning process is the self-realization of students when introducing new organizational forms of using computers into the learning process of computer science and information technology, based on modern packages of symbolic mathematics.

Education within the framework of this experiment allows achieving such goals as self-realization of students and their acquisition of professional competencies, development of mathematical thinking and scientific creativity of schoolchildren, improving the quality and efficiency of the educational process, increasing students' interest in educational activities and interest in its final result, professional orientation students, professional growth of the teaching staff, mastering the methods of information technology, and the creation of computer tools to enhance the educational process.

In the process of studying the Maple Symbolic Mathematics package, students develop practical skills in solving mathematical problems using a computer. Maple becomes their study assistant. Children learn to work on self-control: they solve problems using traditional methods and check the result using Maple. The most interesting and, according to students, useful topics in the program of the elective course were such topics as "Two-dimensional graphics", "Animation", "Research of function". In the process of studying the Maple application, students showed a high cognitive interest and good knowledge of mathematics.

Classes of the elective course are held in various forms: frontal, individual, group. Control and monitoring of students' knowledge, skills and abilities in studying the Maple symbolic mathematics package is carried out in the form of a credit system. During the academic year, students must pass 4 tests in the main sections of the course:

Solving equations, inequalities and their systems;

2D graphics;

Investigation of function and plotting;

Solution of geometric problems.

The final result is the project work of each student. Test papers are issued in the form of Web-documents.

Conclusion

Computer mathematical packages play a very significant role in reforming the teaching of mathematical disciplines in secondary and higher schools, helping to achieve such goals as self-realization of students and their acquisition of professional competencies, the development of mathematical thinking and scientific creativity of schoolchildren, improving the quality and efficiency of the educational process, increasing student interest to educational activity and interest in its final result, professional orientation of students, professional growth of the teaching staff, mastering the methods of information technology, and the creation of computer tools for activating the educational process.

Information support of the educational process is designed to free the student from routine work, allow him to focus on the essence of the material being studied at the moment, consider more examples and solve more problems, facilitate understanding of the material through other ways of presenting the material.

The possibility of computerization of the educational process arises when the functions performed by a person can be formalized and adequately reproduced with the help of technical means. Therefore, before proceeding with the design of the educational process, the teacher must determine the relationship between the parts that can be automated and which cannot.

Multifunctional package Maple is one of the most powerful mathematical packages. Its capabilities cover quite a lot of areas of mathematics and can be usefully applied at different levels, from teaching high school students to the level of serious scientific research. Maple is a system of analytical calculations for mathematical modeling.

The methodology presented in the course work for studying some topics of algebra and starting analysis using the Maple package made it possible to significantly increase the efficiency of the learning process. By visually presenting the material, complex mathematical formulas and transformations become much easier, and the process of assimilation of the material by high school students is much more efficient.

The possibilities of the Maple package, as a means of teaching in high school, are very extensive and its use in the educational process is a promising direction in modern secondary education.

Bibliography

1. Bozhovich, L.I. Personality and its formation in childhood. [Text] / L.I. Bozovic. - St. Petersburg: Peter, 2008. - 398 p.

2. Introduction to Maple. Math package for everyone. V.N.Govorukhin, V.G.Tsibulin, Mir, 1997. - 260 p.

3. Ershov, A.P. School informatics (concepts, state, prospects) / A.P. Ershov, G.A. Zvenigorodsky, Yu.A. Pervin // Informatics and Education.- 1995.- No. 1.- C. 3-19.

4. Lapchik, M.P. Methods of teaching computer science [Text] / M.P. Lapchik, I.G. Semakin, E.K. Hener.- M.: Academy, 2007.- 622 p.

5. Levchenko, I.V. Program and reference materials for teaching practice in informatics: Textbook-methodical. allowance for students ped. Universities and Universities [Text] / I.V. Levchenko, O.Yu. Zaslavskaya, L.M. Dergacheva.- M.: MGPU, 2006.- 123 p.

6. Sdvizhkov, O.A. Mathematics on the Maple 8 computer: Proc. manual for students and university teachers [Text] / O.A. Sdvizhkov.- M.: SOLON-Press, 2003.- 176 p.

7. Semakin, I.G. Informatics. Grade 11: textbook [Text] / I.G. Semakin.- M.: BINOM, Knowledge Laboratory, 2005.- 139 p.: ill.

8. Semakin, I.G. Informatics and ICT. Basic course: textbook for grade 9 [Electronic document] / I.G. Semakin.- (http:www.alleng.ru/edu/comp1.htm). 15.12.08.

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